Diagnostic Test
integration by substitution
"u" substitution or reverse chain rule-- method to find an integral ∫f(g(x))g'(x)dx
∫a^xdx=
(a^x)/lna+C, exponential
ₐ∫^b f(x)dx=
(area above x axis)-(area below x axis)
Chain Rule as a composition of functions f¤g?
(f'¤g)×g'
quotient rule, f/g...?
(gf'-fg')/g²
z score equation
(value-mean)/standard deviation
midpoint formula
(x₁+x₂)/2, (y₁+y₂)/2
slope formula
(y₂- y₁) / (x₂- x₁)
cosπ
-1
Cos(2π/3) or Cos(-4π/3)
-1/2 or ½
what is d/dx (1/x) using the power rule?
-1/x²
derivative of arccosx?
-1/√(1-x^2)
∫sin(x)dx
-cos(x)+C, x is in radians
Reciprocal Rule 1/f
-f'/f²
logarithmic of reciprocal ln(1/x)
-ln(x)
derivative of cosx? x is in radians.
-sinx
cos(3π/4)
-√2/2 Q2
cos(5π/6)
-√3/2 Q2
cos(π/2) or cos(90)
0
derivative of a constant, c?
0
sin0
0
sinπ
0
0!
1
cos 0
1
derivative of a line, x?
1
sin(π/2) or sin(90)
1
event
1 or more outcomes
1 week = _____ days
1 week = 7 days
sin²ϴ=
1-cos²ϴ
16 choose 3, Pascal's triangle requires you to add two numbers together to get the next row, true or false?
1. 14. 91. 364... 1. 15. 105. 405 1365... 1 16. 120. 560. 1820. 4368... 560
see diagram (proofscomp9) given: <1 and <4 are supplementary, prove: a∥b
1. <1 and <4 are supplementary--given 2. <1≅<2 and <3≅<4--are vertical angles- VAT 3. <2 and <3 are supplementary-- substitution property 4. a∥b--converse ssia thm (converse of the same side interior angles theorem)
(see diagram #6) Given: <1≅<4 Prove: <2≅<3
1. <1≅<4 because it is given. 2. <1≅<2---VAT 3. <2≅<4--Transitive Property 4. <3≅<4--VAT 5. <2≅3--Transitive property
Written Indirect Proof - Given: ab=0 and a≠0 Prove: b=0
1. Assume temporarily that b≠0. 2. then ab≠0 since a≠0 and b≠0 because a and b are positive integers. 3. but this contradicts the given statement that ab=0 and a≠0. 4. therefore, the temporary assumption that ab≠0 and a=0 must be false. It then follows that b=0.
If n²>6n, then n≠4
1. Assume temporarily that n=4 2. then n²=16 and 6n=24 due to the multiplication of integers theory. 3. but this contradicts the given fact that n²>6n, since 16 is not greater than 24. 4. therefore the temporary assumption that n=4 must be false. 5. It follows that n≠4.
(see diagram#2) q∥r, r∥s, b⟂q and a⟂s prove: a∥b
1. Because it is given that q∥r and r∥s, then q∥s by the transitive property of parallel lines. 2. <1≅<2 because of the corresponding angles 3. because b⟂q, m<1=90. So the m<2=90°. This means s⟂b, by the definition of perpendicular lines. 4. It is given that a⟂s, so a∥b.
∫(5x+2)⁷dx=
1. Let u=5x+2. Then du=5dx, so ⅕du=dx. 2. Rewrite the problem ∫u⁷×⅕du--- combine= ∫u⁷/5du 3. Since ⅕ is constant with respect to u, move it outside of the integral: ⅕∫u⁷du. 4. By the Power rule, the integral of u⁷ with respect to u is ⅛u⁸, ⅕(⅛u⁸+C)...now combine like terms = u⁸/40+ C 5. Plug 5x+2 back in for u: (5x+2)⁸/40+C
(see diagram #3) Given: g∥h,<1≅<2 Prove: p∥r
1. g∥h -- given 2. <1≅<3-- corresponding angles 3. <1≅<2--given 4. <2≅<3-- transitive property 5. p∥r--converse aea theorem
(see diagram #4) Given: l∥m, a∥b Prove: <1≅<5
1. l∥m-- given 2. <1≅<2 VAT 3. <2 & <3 are supplementary--SSIA 4. <3 & <4 are supplementary--SSIA 5. <2≅<4---≅ supplements thm 6. <1≅<4--transitive property 7. <4≅<5--VAT 8. <1≅<5--transitive property
(Indirect Proof) - if n is an integer and n² is odd, then n is odd.
1. n is even 2. now square n and see what happens. If n is even, then n= 2a, where a is any integer n²=(2a)²=4a² this means that n² is a multiple of 4. No odd number can be divided evenly by an even number, so this contradicts our assumption that n is even. Therefore, n must be odd if n² is odd.
what is ∫e^(x)sin(x)dx?
1. u= sin(x), v=e^(x) 2. differentiate u: sin(x)'= cos(x) 3. integrate v= ∫v= e^(x) sin(x)e^(x)-e^(x)cos(x)-∫e^(x)sin(x)dx now we have the same integral on both sides (except one is subtracted)...so bring the right hand one over to the left and we can get: 2∫e^(x)sin(x)dx=e^(x)sin(x)-e^(x)cos(x)
what is ∫e^(x)dx?
1. u=e^(x) and v=x 2. differentiate u=u'=e^(x) 3. integrate v= ∫xdx=x²/2 e^(x)×x²/2-∫e^(x)×(x²/2)dx
what is ∫e^(x)dx?
1. u=x and v=e^(x) 2. differentiate u=u'=(x)'=1 3. integrate v= ∫e^(x)dx=e^(x)+C xe^(x)-e^(x)+C
what is d/dx sin(x²) using the chain rule?
1. use the chain rule: f(x)=sinx and g(x)=x² 2. f'(g(x))×g'(x) 3. sinx'(x²)×(x²)'...cosx(x²)×2x x²cosx×2x
steps to figure out standard deviation
1. work out the mean 2. then for each number: subtract the mean and square the result (the squared difference). 3. work out the average of those squared differences
(see diagram #5) Given: <1 and <2 are supplementary; x∥y Prove: q∥r
1. x∥y-- given 2. <2&<3 are supplementary-- SSIA thm 3. <1&<2 are supplementary--given 4. <1≅<3--supplements thm 5. q∥r--converse aea thm.
derivative of arctanx?
1/(1+x²)
derivative of logarithm- logₐ(x)?
1/(xln(a))
Sin(π/6) 30
1/2
cos(π/3)
1/2
sin(5π/6)
1/2
median formula
1/2(b1+b2)
∫e^(ax) dx =
1/a integrating the exponential function of course has the opposite effect: it divides by the constant in the exponent.
what is d/dx(1/cosx)?
1/cosx is made up of 1/g and cos(): •f(g)=1/g •g(x)=cos(x) chain rule says the derivative of f(g(x))=f'(g(x))g'(x) The individual derivatives sre: •f(g)=-1/g² •g'(x)=-sin(x) (1/cos(x))'=-1/(g(x))²x-sin(x) =sin(x)/cos²(x) Note: sin(x)/cos²(x) is also tan(x)/cosx, or many other forms.
secϴ
1/cosϴ
what is d/dx (1/x) [use the reciprocal rule]?
1/f=-f¹/f² with f(x)=x, we know that f'(x)=1 so: the derivative of 1/x=-1/x² which is the same result we got using the power rule?
cscϴ
1/sinϴ
cotϴ
1/tanϴ
derivative of logarithm- ln(x)?
1/x
derivative of arcsinx?
1/√(1-x^2)
how many ways can first and second place be awarded to 10 people?
10!(10-2)!=90 P(10,2)
how many permutations of 3 different digits are there chosen from 10 digits 0 to 9 inclusive?
10!/7!= 10×9×8=720
1 liter = ____ milliliters
1000 milliliters
survey a hundred people in our town (7% of our town/population makes up teachers)?
100×7%= 7 teachers
nanogram
10^-9 ng
1cm
10mm
in a lock, there are 10 numbers to choose from (0,1,2,3,4,5,6,7,8,9) and we choose three of them...
10×10×...(3 times)=10³=1000
megameter
10⁶= mm 1×10⁶m
1 foot = ____ inches
12
1 pound = ____ ounces
16 ounces
what order could 16 pool balls be in?
16!= 20,922,789,888,000
how many ways can 3 pool balls be arranged out of 16 balls?
16×15×14= 3360 without repetition our choices get reduced each time.
1 mile is how many yards?
1760 yards
1000 meter are equal to
1km or 1000 millimeters
example: multiply a 1×3 matrix by a 3×1 matrix?
1×1 matrix
kilogram (kg)
1×10³ g= 1 kg or 1000 grams
1 pint = ____ cups
2 cups
1 ton = ____ pounds
2,000 pounds
1 day = _____ hours
24 hours
derivative of a square, x²?
2x
#9- which of the following graphs represents the solution set of the inequality 2(x-3)<4(x+1)?
2x-6<4x+4 x>-5
360° in radians
2π or approximately 6.283 radians
find the period of the function -2cos(3x)?
2π/|b|= 3x=2π= x= ⅔π
find the period of the function 3sin(4x)?
2π/|b|= 4x=2π= x= ½π
find the period of the function -2cosϴ/6?
2π/|b|= ϴ/6=2π= ϴ = 12π
1 yard = ____ feet
3 feet
1 year
365 Days/52 weeks/12 months
what is d/dx (x³) using the power rule?
3x²
example: multiply a 3×1 matrix by a 1×3 matrix?
3×3 matrix
270° in radians
3π/2 or 4.712 radians
1 gallon = ____ quarts
4 quarts
If there are 45 numbers, what is the middle number?
45 plus 1 is 46, then divide by 2 and we get 23.
#32- five different algebra textbooks, two different calculus textbooks, in four different geometry textbooks are to be arranged on the Shelf. How many different arrangements are possible if the books of each subject must be kept together?
5!2!4!3! this question requires examity to apply concepts of permutations and combinations to solve problems. If the books in each of the three subjects must be kept togethe, then the number ways the groups of books can be arranged by subject is represented by 3 factorial. if there are n books within a subject, the number of ways the books can be arranged is n!. thus the algebra books can be arranged in 5! different ways, the calculus books can be arranged 2 factorial different ways and the geometry books can be arranged in 4! different ways. Since there is Independence between the different Arrangements computed, the total number of ways the books can be arranged is a product of all factorials 3 factorial 5 factorial 2 factorial 4 factorial which is equivalent to 5 factorial 2 factorial 4 factorial 3 factorial. review competency 16 for more practice
1 mile = ____ feet
5,280 feet
1 radian
57.3 degrees or 180/π
1 hour = _____ minutes
60 minutes
1 minute = _____ seconds
60 seconds
If there are 66 numbers, what is the middle number?
66 plus 1 is 67, then divide by 2 and we get 33.5. That means the 33rd and 34th numbers in the sorted list are the two middle numbers.
a password consists of two letters of the alphabet followed by 3 digits chosen from 0 to 9. repeats are allowed. how many different possibly passwords are there?
676,000 The # of ways of choosing the letters =26×26=676 The # of ways of choosing the digits =10×10×10=1000 Using the basic counting principle =676×1000= 676,000
standard deviation 68%
68% of values are within 1 standard deviation of the mean
1 cup= ? oz
8 oz
standard deviation 95%
95% of values are within 2 standard deviations of the mean
standard deviation 99.7%
99.7% of values are within 3 standard deviations of the mean
what is ∫8z+4z³-6z²dx (use the sum and difference rule)?
= ∫8zdz+∫4z³dz-∫6z²dz Constant Multiplication =8∫zdz+4∫z³dz-6∫z²dz Power Rule =8z²/2+4z⁴/4-6z³/3+C simplify =4z²+z⁴-2z³+C
∫cos(x²)6xdx=
=3∫cos(x²)2xdx Then go ahead as before: 3∫cos(u)du=3sin(u)+C Now put u=x² back again: 3sin(x²)+C
what is ∫e^(w)-3dw(use the difference rule)?
=∫e^(w)dw-∫3dw =e^(w)-3w+C
Histogram
A graph of vertical bars representing the frequency distribution of a set of data.
centimeter (cm)
A metric unit of length equal to 0.01 of a meter. 1×10^(-2)m
kilometer (km)
A metric unit of length equal to 1000 meters.
square (need to see diagram?)
A parallelogram with four congruent sides (all equal) and four right angles. (also a rectangle and rhombus)
rectangle
A parallelogram with four right angles (has 2 pairs of of equal opposite parallel sides)
outcome
A possible result of a probability experiment
Trapezoid
A quadrilateral with exactly one pair of parallel sides ( 1 pair of opposite sides....parallel lines)...(UK: Trapezium-- has no parallel lines in US and has a pair of parallel lines in UK.
Vector
A quantity that has magnitude (size) and direction
Vector
A quantity that has magnitude and direction
identity matrix
A square matrix with ones (1s) along the main diagonal, from the upper left element to the lower right element, and zeros (0s) everywhere else. Symbol is I.
event
A subset of a sample space/ 1 outcome ie. getting a tail when tossing a coin, rolling a "5"
truth table
A table used as a convenient method for organizing the truth values of statements
isosceles triangle
A triangle that has 2 equal or congruent sides.
dividing matrices
A/B=A×(1/B)=A×B^(-1) we actually don't divide but have an inverse
given the system: x +8y=7 2x-8y=-3, the coefficient matrix is:
A=[1. 8] [2.-8]
#16- the shape of B consists of rectangles and semicircles. If the space except for the two openings are shaded in, what formula gives the A for area?
A=h²(1/6+π/18) this question requires examinee to apply the concepts of perimeter, circumference, area, surface area, and volume to solve real-world problems. The total area of the letter B can be viewed as the area of an h x h/6 rectangle plus the area of a circle with radius h/12 or h²/6 or π(h/4)²-π(h/12)² this simplifies h²/6+πh²/16-πh²/144 and further to h²(1/6+π/16-π/144) and further to h²(1/6+9π/144-π/144) and h²(1/6+π/18).
#31- Matrix a has Dimensions 4 by 2 Matrix B has dimensions 4x 2 and Matrix C has Dimensions 2 by 4. Which of the following operations is defined?
ACB, this question requires the examinee to perform an operation on matrices. Matrix a has 4 rows and 2 columns; Matrix B has 4 rows and 2 columns and matrix c has two rows and 4 columns. Matrix multiplication requires that the number of columns in the first factor X can be the same as a number of rows in the second Factor of y, yielding the matrix product XY with the number of rows as in the X and in the number of columns as in y. Applying these rules (4x 2) (2x4) (4x2) yields the defined 4 by 2 Matrix ACB. review competency 16 for more practice.
permutation
An arrangement, or listing, of objects in which order is important. aka ordered combination
what is ₁∫²2xdx?
At x=1, : ∫2xdx=1²+C At x=2, :∫2xdx=2²+C subtract: (2²+C)-(1²+C) 4-1=3 and C gets canceled out, so with definite integrals we ignore C.
special matrix
A×I=A I×A=A
given the system: x +8y=7 2x-8y=-3, the constant matrix is:
B= |7| |-3|
#11- the loudness L and the intensity l of a sound are related by the equation L = 10 log I/I⁰ where L is measured in decibels (dB) and L is measured in watts p/square meter (W/m²) where Iₒ=10^(-12) W/m² is the intensity of a barely audible sound. what is the intensity of a 90 decibel sound made be a subway train?
Competency 6 analyze real-world problems involving exponential and logarithmic functions. Substituting the given values into the equation L=10*log (I/I₀) 90=10*log(I/10^(-12)) [(divide both sides by I/10^(-12))] I/10^-12 = 10⁹ I=10^(-12) * 10⁹ [multiply by 10^(-12) on both sides] I = 10^(-3) I=10^(-3) W/m² more practice in Chapter 6
dependent events
Events for which the outcome of one event affects the probability of the second event
mutually exclusive
Events that cannot occur at the same time.
(Indirect proof) Prove SSS inequality theorem is true by contradiction. (The SSS inequality theorem says: if two sides of a triangle are congruent to two sides of another triangle but the third side of the first triangle is longer than the third side of the second triangle and the included angle of the first triangle has two congruent sides is greater and measure than the included angle of the second triangle has two congruent sides)
First, assume the opposite of the conclusion. The included angle of the first triangle is less than or equal to the included angle of the second triangle. If the included angles are equal then the two triangles would be congruent by SAS and the third sides would be congruent by cpctc. This contradicts the hypothesis of the original statement that "third side of the first triangle is longer than the third side of the second" Therefore, the included angle of the first triangle must be larger than the included angle of the second.
Reflexive Property of Equality
For any real number a, a=a
tossing a coin three times
HHH, HTT, TTH, HHT, THH, TTT, HTH, THT
ILATE
I-inverse trigonometric functions such as sin^(-1)(x), cos^(-1)(x), and tan^(-1)(x) L-logarithmic functions such as Ln(X), log X A- algebraic functions such as x², x³ T-trigonometric functions such as sin(x), cos(x), tan(x) E-exponential functions such as e^(x), 3^(x)
Base Angles Converse Theorem
If 2 angles of a triangle are congruent, then the sides opposite them are congruent
Matrix Multiplication
If 3x3 and 3x3: Multiply row 1 of matrix A by column 1 of matrix B, row 1 by column 2, row 1 by column 3. Congrats! You got your first row. Repeat with rows 2,3. This is [AB]. Matrix multiplication are not commutative. # columns (n) in first matrix = # rows (m) in second matrix matrix multiplication requires that the number of columns in the first factor A be the same as a number of rows in the second factor of B yielding the matrix product A* B with a number of rows as in A in the number of columns as in B.
Same-Side Interior Angles Theorem
If a transversal intersects two parallel lines, then same-side interior angles are supplementary.
Transitive Property
If a=b and b=c, then a=c
Substitution Property
If a=b, then a can be substituted for b in any equation or expression
Congruent Supplements Theorem
If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.
Vertical Angles Theorem
If two angles are vertical angles, then they are congruent.
Converse of SSIA (same side interior angles) Theorem
If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel.
Converse of AEA Theorem
If two lines are cut by a transversal forming congruent alternate exterior angles, then the lines are parallel
transitive property of parallel lines
If two lines are parallel to the same line, then they are parallel to each other. ie.: if line E is ∥to line F is ∥to line G, then line E is ∥to line G
CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
If two triangles are congruent, then their corresponding parts are congruent to each other. Used as a justification in a proof after you have proven two triangles congruent. If you know that two triangles are congruent (SSS, SAS, ASA, AAS), then you know that ALL corresponding parts are congruent.
perpendicular lines
Lines that intersect to form right angles
dot product
Multiplying two vectors by multiplying parallel components. Result is a scalar. v=<a,b> and u=<c,d> v•u=ac+bd to find the angle between the vectors: cosϴ=v/|v|•u/|u|
similarity
Objects that are similar in appearance are more likely to be perceived as belonging in the same group.
P(Two Heads or x=2)
P(HHT)+P(HTH)+P(THH)= ⅛+⅛+⅛= 3/8
P(One Head or x=1)=
P(HTT)+P(THT)+P(TTH)= ⅛+⅛+⅛=3/8
P(Zero Heads or x=0)
P(TTT)= ⅛
∫1/x dx
Reciprocal ln|x| + C
#22- which of the following are solutions to 2sin²ϴ=cosϴ+1 for 0<ϴ<2π?
Replace the 2sin²ϴ with 2(1-cos²(ϴ)) based on the sin²ϴ+cos²ϴ=1 identity 2-2cos²ϴ-cosϴ-1=0 -2cos²ϴ-cosϴ+1=0 (2cosϴ-1)(-cosϴ-1)=0 1. ϴ=arccos(-1)=π 2. ϴ=arccos(½)=π/3 1a. the cosine function is negative in the 2nd and 3rd quadrants. to find the second solution, subtract the reference angle from 2π to find the solution in the third quadrant. ϴ=2π-π=π 1b. Find the period. 2π/1=2π *The period of the cos(ϴ), so the values repeat every 2π radians in both directions. ϴ=π+2πn, for any integer, n. 2. the cosine function is positive in the first and four quadrants. To find the second solution subtract the reference angle from 2π, to find a solution in the fourth quadrant. ϴ=2π-π/3=5π/3 ϴ=π+2πₙ, π/3+2πₙ, 5π/3+2πₙ, for any integer n.
Gaussian elimination
Row-reduce until you get an x = number and substitute until you get an actual number.
Trigonometric Functions
SOH CAH TOA (sine is opposite over hypotenuse, etc.)
opposite angle identities
Sin(-A)=-sinA Cos(-A)=CosA tan(-A)=-tanA
~P∧(P->Q) truth table
TTFTF TFFFF FTTTT FFTTT
P∧Q (P and Q) Truth Table
TTT TFF FTF FFF P∧Q should be true when both p and Q are true and false otherwise
P<->Q
TTT TFF FTF FFT p and Q are equivalent. so the double implication is true if both p&q or true or both false
P->Q
TTT TFF FTT FFT table for logical implication if-then... statement will be true if I keep my promise and false if I don't
PVQ (P or Q) Truth Table
TTT TFT FTT FFF PVQ is true if either p is true or Q is true (or both). It's only false if both p and Q are false.
notation for matrix
The Matrix is usually shown by a capital letter such as (A or B). each entry ("or element") is shown by a lowercase letter with a "subscript" row, column: A= [a₁.₁ a₁.₂ a₁.₃] [a₂.₁ a₂.₂ a₂.₃]
angle of depression
The angle formed by a horizontal line and the line of sight to an object below the horizontal line
mean formula
The arithmetic average. To find the mean (x bar) of a set of observations, add their values and divide by the number of observations.
interquartile range
The difference between the upper and lower quartiles.
binomial coefficient
The number of ways of arranging r success in n trials. where n is the # of things to choose from, and we choose r of them, no repetition, order doesn't matter n!/r!(n-r)! for combinations C(n,r)
Data Analysis
The process of compiling, analyzing, and interpreting the results of primary and secondary data collection to help make decisions
Ground Speed
The speed of the aircraft in relation to the speed on the ground.
Air Speed
The velocity of the aircraft
mode formula
To find the mode, or modal value, it is best to put the numbers in order. Then count how many of each number. A number that appears most often is the mode.
row equivalent
Two matrices are row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other.
Quartiles
Values that divide a data set into four equal parts
given the system: x +8y=7 2x-8y=-3, the variable matrix is:
X=|x| |y|
[2 0] [1 2] [1 2] [3 4]?
[2×1+0×3. 2×2+0×4] [1×1+2×3. 1×2+ 2×4] [2. 7] [7. 10]
[1 23] x. [7 8] [456] [9 10] = [11 12]
[58. 64] [139 154] (1,2,3)×(7,9,11)=1×7+2×9+3×11=58 (4,5,6)×(8,10, 12)=8×4+5×10+6×12+=154
normal distribution
a bell-shaped curve, describing the spread of a characteristic throughout a population • includes mean, median, and mode •has a line of symmetry, 50% of values less than mean...50% of values greater than mean
tree diagram (probability)
a diagram used to show the total number of possible outcomes in an experiment (see diagram)
quadrilateral
a four-sided polygon (quad means 4, lateral means side). properties: •polygons •four sides (edges) •four vertices (corners) •interior angles that add up to 360°
sinusoidal function
a function that is like a sine function in the sense that the function can be produced by shifting, stretching or compressing the sine function
combination
a grouping of items in which order does not matter 2 types: repetition (such as coins in your pocket- 5,5,5,10,10) and no repetition [such as lottery numbers (2,14,15,27,30,33)]
Transversal
a line that intersects two or more lines
standard deviation
a measure of variability that describes an average distance of every score from the mean --(similar symbol)- σ=√(variance)= σ=√[np(1-p)]
Rhombus
a parallelogram with 4 congruent sides (4 equal opposite angles are equal) and diagonals that bisect at right angles aka quadrilateral, rhomb or diamond.
Parallelogram
a quadrilateral with both pairs of opposite sides parallel
Matrix
a rectangular arrangement of numbers or terms having various uses such as transforming coordinates in Geometry, solving systems of linear equations in linear algebra and representing graphs in graph Theory
experiment
a repeatable procedure with a set of possible results that is used to test a hypothesis
random sampling
a sample that fairly represents a population because each member has an equal chance of inclusion need a full list of a population to pull from
cluster sampling
a sampling technique in which clusters of participants that represent the population are used example, we divide the town into many different zones, and randomly choose five zones and survey every one of those zones.
median of a triangle
a segment from a vertex to the midpoint of the opposite side
Loudness
a sound's intensity L=10log(I/Iₒ)
contradiction
a statement that is the opposite of another statement
isosceles trapezoid
a trapezoid with only congruent legs
scalene triangle
a triangle with no congruent or equal sides
stratified sampling
a variation of random sampling; the population is divided into subgroups and weighted based on demographic characteristics of the national population (make each group is proportional to those represented in the population)
unit vector
a vector that has a magnitude of 1 unit v/|v|
vector sum
a vector that is the sum of two or more other vectors
Amplitude of a trig function
absolute value of a
adding two matrices
add the numbers in the matching positions
sample space of 52 cards:
all of the cards including the jokers {ace of hearts, 2 of hearts, etc....}
Matrix
an array of numbers
trigonometric identity
an equation involving a trionometric ratio that is true for all values of the angle measure: sin, cos, tan
independent event (probability)
an event that is not affected by another event.
definite integral
an integral expressed as the difference between the values of the integral at specified upper (a) and lower limits (b) of the independent variable.
alternate interior angles
angles between 2 lines and on opposite sides of a transversal are congruent
alternate exterior angles are
are angles on the opposite sides of the transversal and opposite the two lines. If the two lines are parallel, then the angles are congruent.
(Indirect proof) if <A and <B are complimentary, then <A≤90°. Prove this by contradiction.
assume the opposite of the conclusion. Angle A> 90° consider first that the measure of < B cannot be negative. So if <A >90° this contradicts the definition of complementary, which says that two <'s are complementary if they add up to 90°. therefore, <A < 90°.
mean
average
Pythagorean Theorem
a²+b²=c²
law of cosines formula
a²=b²+c²-2bcCosA b²=a²+c²-2acCosB c²=a²+b²-2abCosC Used for finding: • the third side of a triangle when we know two sides and the angle between them • the angles of a triangle when we know all three sides
2. in four 1/2 cup samples of cereal containing dried cranberries, the numbers of cranberries were 17, 22, 22 and 18. Nutrition information on a box of this cereal defines the serving size as 1 cup or 50 grams. If a box contains 405 grams, which of the following is the best estimate of the number of cranberries in one box of cereal?
between 300 and 325 competency 1- use estimation (i. E., Rounding, area and plausibility). There are approximately 20 cranberries per half a cup or 40 cranberries per cup. The number of cups in a box is 405 divided by 53 grams which when rounded is 8. the approximate number of cranberries in a box 8 * 40 equals 320. (chapter one has extra practice)
quincunx
board developed by Sir Francis Galton to demonstrate bell-shaped curve triangular array of pegs
what is ∫ln(x)dx?
but there is only one function! how do we choose u and v. hey, we can just choose V as being "1". 1. u=ln(x) v=1 2. Differentiate u'=1/x ∫1= x+C Simplify =xln(x)-∫1dx =xln(x)-x+C
derivative of cf? (multiplication by constant)
cf'
vector subtraction
change direction of subtracted vector then vector addition u+(-v)
vector multiplication by scalar
change magnitude, may reverse direction kv=(ka,kb)
choosing u and v
choose a "u" that gets simpler when you differentiate it and a "v" that doesn't get any more complicated when you integrate it
n×n...(r times) or n
choosing "r" of something that has "n" different types is permutation...
#1 there was a mixture of red, blue and yellow marbles in a bag. The total number of marbles is 70 and there are 6 times as many blue marbles as yellow marbles in the bag? how many possible solutions are there for the problem above?
competency 1 - this question requires the examinee to solve mathematical in real-world world problems involving integers, fractions, decimals and percents. Since there are 6 times as many blue marbles, B as yellow marbles, Y in the bag, b=6Y and the blue and yellow marbles could be sorted evenly into groups of 7 marbles each containing six blue marbles and one yellow marble. The possible number of red marbles can be represented as 70 - 7n. Where n represents the number of groups of Blue and yellow marbles. Given that there is at least one red marble in the bag, the possible integer values for n are 1 -9, each representing a different solution to the problem. (additional practice in chapter 1)
#12- which of the following is equivalent to the equation 3log₁₀10X-2log₁₀10Y=17?
competency 6 this question requires the examinee to apply the laws of exponents and logarithms. 1. Power rule: NlogₐM=logₐM^(N)=> 2.3logₐ10X=log₁₀X³ and 2log₁₀Y=log₁₀y² 3. Quotient Rule: LogₐM-logₐN=> log^(m/n)=>log₁₀X³-log₁₀Y²=log₁₀x³/y² since logₐM=N is equivalent to a^(N)=M, then log₁₀x³/y²=17 is equivalent to 10¹⁷=x³/y² review chapter 6 for more practice
which of the following could be the graph of the function f(x)= a√b(x - c) where a< 0 B<0 and C<0?
competency 7 this question requires examinee to analyze the relationship between a radical function and its graph. Changing the leading coefficient in an equation from positive to negative reflects the graph of the equation over the x-axis. In this case since a<0, the graph of the function in the correct response must be reflection of f(x)=|a|√b(x-c) over the x-axis.
#17- see diagram given ray AC bisects < BAD and < BCA is acute, prove: AB≠AD
competency 9 this question requires the examinee to analyze formal and informal geometric proofs. In an indirect proof, assume that the conclusion is not true and then apply reason until the hypothesis or known fact is contradicted. Here the desired conclusion is AB≠CD, so assume AB=AD.
logical connector: ∧
conjunction...and
∫a dx
constant ax+C
parallel lines
coplanar lines that do not intersect
Congruence
correspondence of parts; harmonious relationship; CF. congruity
Sum Identities
cos(A+B) = cosAcosB - sinAsinB sin(A+B)= sinAcosB+cosAsinB tan(A+B)=tanA+tanB/1-tanAtanB
the difference identities
cos(A-B) = cosAcosB + sinAsinB sin(A-B)= sinAcosB-cosAsinB tan(A-B)=tanA-tanB/1-tanAtanB
derivative of sinx? x is in radians.
cosx
Periodic Function: cos(2nπ+ϴ) or cos(2π+ϴ)=
cosϴ, periodic function of 2π function f(x) is periodic in trigonometry if there exists a real number T>0 such that f(x+T)=f(x) for all x. If T is the smallest positive real # such that f(X+T)=f(x) for all x then T is the fundamental period of f(x).
vector components directional angle
cosϴ=x/|v| and sinϴ=y/|v|
Distance Formula
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
what is d/dv(v³-v⁴) [use the difference rule]?
difference rule says: f'-g' 3v²-4v³
d/dxe(a^x)
differentiating the exponential function just multiplies it by the constant in the exponent
logical connector: ∨
disjunction...or
tan²ϴ+1=
dividing by cos²ϴ gives us: sin²ϴ/cos²ϴ+1= 1/cos²ϴ= tan²ϴ+1=sec²ϴ
1+cot²ϴ=
dividing by sin²ϴ gives us: cos²ϴ/sin²ϴ+1= 1/sin²ϴ= so, cot²ϴ+1=csc²ϴ
Chain Rule (dy/dx)
dy/du * du/dx
derivative of exponential, e^(x)?
e^(x)
∫e^xdx=
e^x+C, exponential
Law of the Exceeded Middle
every statement is either true or false.
Chain Rule f(g(x))
f'(g(x)) x g'(x)
Sum Rule (f+g)
f'+g'
difference rule, f-g?
f'-g'
#4- given statements p and Q which of the following is a truth table for the compound statement p<->(qv~p)?
for ~p is computed as fftt. finally this result is used to compute truth values for the full expression p<->(qv~p): T,F,F,F...anything else to review... *P∨Q is true if either P or Q is true. It is only false if P or Q is false. *P<->Q means P and Q are equivalent. So the double implication is true if both p&q or true or both false.
tautology
formula that is only or always true
product rule, f×g?
f×g'+g×f'
matrix rows
go from left to right, come before columns
Matrix columns
go up and down
AX=b
if a is a m×n Matrix and x designates a column Vector (IE. n x 1 Matrix) of n variables X₁X₂,...Xₙ and b is an annuity is an m x 1 column Vector, then the Matrix equation is AX=b
<->
if and only if
an encyclopedia has eight volume. In how many ways can 8 volumes be replaced on the Shelf?
imagine there are eight spots on the Shelf. Replaced the volumes one by one. The first volume to replace to go in any one of the eight spots. The second volume to be replaced then go in any one of the seven remaining spots. The third volume to be replaced could then go in any one of the six remaining spots. 8!=40320
throw the dice - a fair die is thrown four times. Calculate the probabilities of getting 0- twos, 1- two, 2- twos, 3- twos, 4- twos...
in this case, n=4, P(two)=⅙, x is the random variable...number of twos from four throws... p(x=0)= 4!/0!4!×⅙×(⅚)⁴=1×1×(⅚)⁴=.4823 p(x=1)=4!/1!4!×(⅙)×(⅚)³=4×(⅙)×(⅚)³=.3858 p(x=2)=4!/2!4!×(⅙)²×(⅚)²=6×(⅙)²×(⅚)²=.1157 p(x=3)=4!/1!(3!)×(⅙)³×(⅚)¹=4×1/216×⅚=20/1296=.0154 p(x=4)=4!/4!(0!)×(⅙)⁴(⅚)⁰=1/1296=.0008
meter
instrument for measuring, 100cm
periodic function
is a function for which a specific horizontal shift p results in the original function: f(x+p)= f(x) for all values of x. When this occurs, we call the smallest such horizontal shift p>0, the period of the function.
sample point
just one of the possible outcomes
Bias
land more on one side than another or choices that are not 50/50
corresponding angles
lie on the same side of the transversal and in corresponding positions
Probability
likelihood that a particular event will occur
Limit of difference quotient
limit as h approaches 0 f(1+h)-f(1)/h
log of 1
ln(1)=0
derivative of exponential, a^(x)?
ln(a)a^(x)
log of e
ln(e)=1
power rule of logarithms
logb Mp = p · logb M, with M, N , b > 0, b ≠ 1.
Logarithmic Power Property
logb(X^y)=ylogb(X)
Quotient of Logs
logb(x/y)=logb(x)-logb(y)
Logarithm of a Product
logb(xy) = logb(x) + logb(y)
length of a vector
magnitude |v|=√(a²+b²) tied to Pythagorean Theorem
95% of students at school are between 1.1m and 1.7m tall. calculate the mean and standard deviation?
mean= (1.1m-1.7m)/2=1.4m 95% is 2 standard deviations on either side of the mean ( a total of 4 standard deviations) so: 1 standard deviation = (1.7m- 1.1m)/4 =.6m/4 =.15m
micrometer
millionth of a meter 1×10^(-6)mg
toss a coin 100 hundred times, how many heads will come up?
most cases 50% chance but when we actually try it we might get 48 heads, or 55 heads....or anything really
what is d/dx 5x³ (hint: multiplication by constant)?
multiplication by constant 15x²
scalar multiplication of a matrix
multiplying any matrix by a constant called a scalar; the product of a scalar k and an m x n matrix
aces and kings: mutually exclusive or not mutually exclusive...?
mutually exclusive, can't be both
Matrix Multiplication
m×n X n×p ->m×p
n choose k formula
n! / (k!)(n-k)! •n= total # •k=number we want
Permutation Formula
n!/(n-r)!
general binomial probability formula
n!/k!(n-k)!p^(k)(1-p)^(n-k)
with 3 tosses, what are the chances of 2 heads?
n=3, k=2 =3!/2!(3-2)! =3 so there are 3 outcomes with 2 heads
with 9 tosses, what are the chances of 5 heads?
n=9, k=5 =9!/5!4! =126 **and for 9 tosses there are a total of 2⁹=512 outcomes, so we get the probability: number of outcomes we want: 126, probability of each outcome: 1/512 =126/512= about a 25% chance
permutation notation
nPr=n!/(n-r)!; ^nPᵣ
Pascal's Triangle
named after Blaise Pascal-- a famous French mathematician and philosopher. an arrangement of the values of nCr in a triangular pattern where each row corresponds to a value of n horizontal sums: 11⁰=1 11¹=11 11²=121 11³=1331 11⁴=14641 or 1st diagonal- ones 2nd diagonal- counting numbers 3rd diagonal- triangular numbers 4th diagonal- tetrahedral #'s: 1,4,10,20,35
logical connectors, ~
not
~
not
hearts: mutually exclusive or not mutually exclusive...?
not mutually exclusive, can be both
probability of an event happening:
number of ways it can happen/ (divided by) total # of outcomes
There are 5 marbles in a bag. Four are blue and one is red. What is the probability that a Blue Marble gets picked?
number of ways it can happen: 4 total # of outcomes: 5 marbles in total so the probability = ⅘=.80
Power Rule (x^n)
nx^(n-1)
Power Rule of Integration, ∫x^(n) dx
n≠1, ∫[x^(n+1)/n+1]+C
continuous
oh yes the function we are integrating must be continuous between a and b: no holes or vertical asymptotes (where the function heads up/down towards Infinity)
milligram (mg)
one thousandth of a gram (0.001 g) 1×10^(-3)g
#7- If f(x)=x^2 and g(x)=x-2, what is the value of (f+g)(2)-(f-g)(3)?
operations with functions (f+g)(2) represents the sum of f(2) and g(2) and (f-g)(3) represents the difference of f(3)-g(3). Thus (f+g)(2)-(f-g)(3)=[f(2)+g(2)]-[f(3)-g(3)]=[2^2+(2-2)]-[3^2-(3-2)]=[4+0]-[9-1]=4-8=-4
Negation Truth Table
p ~p T F F T easy... if p is true, its negation ~p is false. if p is false, then ~p is true.
p (3 heads or x=3)
p(HHH)= ⅛
Coplanar
points that lie on the same plane
Sigma
population standard deviation
what is the derivative of cos(x)sin(x) [use the product rule]?
product rule= f•g= fg'+gf' cosxcosx+-sinxsinx =cos²x-sin²x
without replacement
put the selected object aside before drawing the next, the events are dependent
#3- given the horizontal and vertical components of a vector, which of the following is a basis for calculation of the magnitude of a given vector?
pythagorean theorem i.e., absolute value of a = square root of x²+y² review chapter 2 exercises if needed
what is the derivative of cos(x)/x [use the quotient rule]?
quotient rule= (f/g)'= f'g-fg'/g² f=cosx g=x (cosx)'•x-cosx(x)'/(x)² sinx(x)-cosx•1/x=[x(sinx)+cosx]/x²
(Algebra example indirect proof) If x=2, then 3x-5≠10. prove the statement is true by contradiction.
remember that in an indirect proof the first thing you do is assume the conclusion of the statement is false. In this case, we will assume the opposite of "if x=2, then 3x-5≠10 take the statement is true and solve for x 3x-5=10 3x=15 x=5 but x=5 contradicts the given statement that x= 2. Hence, our assumption is incorrect and 3x - 5≠ 10 is true.
(Indirect proof) If triangle ABC is isosceles, then the measure of the base angles cannot be 92°. Prove this indirectly.
remember, to start assume the opposite of the conclusion. The measure of the base angles are 92°. If the base angles are 92,° and they add up to 184°. This contradicts the Triangle Sum Theorem that says the three angle measures of all triangles add up to 180 degrees. therefore, the base angles cannot be 92 degrees.
ₐ∫^b f(x)dx=-ᵦ∫^ɑ f(x)dx
reversing the direction of the interval gives the negative of the original Direction
#29 - a store is randomly giving away 1 of 12 different gifts to each shopper during its grand opening. Which of the following simulations would best estimate the probability of a shopper receiving a particular gift?
roll a standard die, flip a standard coin and record the outcome. this question requires the examinee to select a simulation that models a real-world event. A standard die has six sides numbered 1 through 6 and a standard coin has two sides, heads (H) and tails (T). when a coin is flipped and a die is rolled, there are 12 possible outcomes. H1T1, H2T2, H3T3, H4T4,H5T5, H6 AND T6 of which has a probability that the shoppers will receive a particular gift of 12 possible gifts. review competency 15 if needed.
Alex wants to see how many times a double comes up when throwing two dice?
sample space - all possible outcomes (36 sample points) {1,1} {1,2} {1,3} {1,4}...{6,3}{6,4}{6,5}{6,6} double Alex is looking for is: {1,1}{2,2}{3,3}{4,4}{5,5} and {6,6}...6 sample points After 100 experiments, Alex has 19 double events...is this what you would expect?
derivative of tanx? x is in radians.
sec²x
∫cos(x) dx=
sin(x)+C, x in radians
supplementary angle identities
sinA= sin(pπ-A) where 0<= A<= pπ
trigonometric identity reciprocal
sinA=1/cscA cosA=1/secA tanA=1/cotA cscA=1/sinA secA=1/cosA cotA=1/tanA
complementary angle identities
sin[(π/2)- ϴ]=cosϴ tan[(π/2) -ϴ]=cotϴ sec[(π/2)-ϴ]=cscϴ
prove that tanx+cotx=secxcscx
sinx/cosx+cosx/sinx find a common denominator-- =(sin²x+cos²x)/cosxsinx=1/cosxsinx =secxcscx
prove that sinxcosxtanx=1-cos²x
sinxcosx×sinx/cosx (sin²x×cosx)/cosx-- divide by cosx sin²x=1-cos²x
prove that tany/siny= secy
siny/cosy×1/siny= cancel out siny 1/cosy= secy
Pythagorean Identities
sin²x+cos²x=1 1+tan²x=sec²x 1+cot²x=csc²x
Periodic Function: sin(2nπ+ϴ) or sin(2π+ϴ)=
sinϴ, periodic function of 2π function f(x) is periodic in trigonometry if there exists a real number T>0 such that f(x+T)=f(x) for all x. If T is the smallest positive real # such that f(X+T)=f(x) for all x then T is the fundamental period of f(x).
Gauss-Jordan Elimination
solving a system of linear equations by transforming an augmented matrix so that it is in reduced row-echelon form
you want to know the favorite colors for people at your school, but don't have time to ask everyone.
somehow get a full list of students printed out, then •place all the pages on the ground, drop a pencil and note down the student's name repeat until you have 50 names.
Variance
standard deviation squared, σ²=np(1-p)
to subtract 2 matrices:
subtract the numbers in the matching positions Note: subtracting is actually defined as the addition of a negative Matrix: A+(-B)
what is the derivative of x²+x³ (use the sum rule)?
sum rule: f'+g'= 2x+3x²
random variable, x
takes numerical values that describe the outcomes of some chance process or experiment
∫sec²(x) dx=
tan(x)+C, x is in radians
trig shadow problem (see diagram) the length of a tree Shadow is 20 feet. when the angle of elevation to the sun is 40°. How tall is the tree?
tan40°=y/20 tan40°×20=y .8390×20=y 16.78ft=y
Ratio Identities
tanθ = sinθ/cosθ cotθ = cosθ/sinθ
Periodic Function: tan(π+ϴ) and cot(π+ϴ)=
tanϴ and cotϴ, periodic functions of π function f(x) is periodic in trigonometry if there exists a real number T>0 such that f(x+T)=f(x) for all x. If T is the smallest positive real # such that f(X+T)=f(x) for all x then T is the fundamental period of f(x).
direction of a position vector
tanϴ=b/a
derivative
tells us the slope of a function at any point.
16 teams enter a competition. They are divided up into four pools (A, B, C and D) of four teams each. Every team plays one match against the other teams in its pool. After the pool matches are completed: • the winner of pool A plays the second-placed team of pool B • the winner of pool B plays the 2nd placed team of pool A •the winner of pool C plays the second-placed team of pool D •the winner of pool D plays the second-placed team of pool C. the winners of these four matches then play semi-finals, and the winners of the semi-finals play in the final. how many matches are played together?
the # of matches played in each pool is "4 choose 2" = ⁴C₂= 4!/(2!2!)= (4×3)/2×1=6 Imagine 4 teams (A,B,C,D) A plays B A plays C A plays D B plays C B plays D C plays D They each play eachother 6 games in total. With 4 pools, the total # of pool matches = 4×6= 24 The winners and 2nd placed teams play a further 4 matches. Then there are 2 semi-finals and 1 final. 24+4+2+1=31
assuming that any arrangement of letters form a word, how many 'words' of any length can be formed from the letters of the word SQUARE?
the # of one letter 'words' = ⁶P₁, 6 the # of two letter 'words' = ⁶P₂, 6×5=30 the # of 3 letter 'words' = ⁶P₃, 6×5×4=120 the # of 4 letter 'words' = ⁶P₄, 6×5×4×3=360 the # of 5 letter 'words' = ⁶P₅, 6×5×4×3×2=720 the # of 6 letter 'words' = ⁶P₆=6!=720 so the total # of possible words =6+30+120+360+720+720=1956
vector addition
the adding or combining of vector magnitudes and directions
amplitude of a sinusoidal function2
the amplitude of the sine and cosine functions is a vertical distance between the sinusoidal access in the maximum and minimum value of the function. f(x): ±asin(b(x+c))+d a=amplitude |a| b=period= 2π/|b| c= phase shift= c/b d= vertical shift= d
angle of elevation
the angle formed by a horizontal line and the line of sight to an object above the horizontal line
position vector
the arrow on a motion diagram that is drawn from the origin to the moving object (a,b) or terminal point. CD(On top of CD is -->)= (x₂-x₁, y₂-x₁)=(a,b) ⃤x, ⃤y
Gambler's Fallacy
the belief that the odds of a chance event increase if the event hasn't occurred recently
example: you're off to a soccer, and and love being a goalkeeper, but that depends on who is the coach today. • with coach Sam, your probability of being goalkeeper is .5 •with Coach Alex your probability being goalkeeper is .3
the branches for Sam (.5 Yes and .5 No), and then for Alex (.3 Yes and .7 No)
example: drawing two cards from a deck after taking one card from the deck there are less cards available so the probabilities change!
the chances of getting a king are = 4/52 •if the first card was a king, then the second card is less likely to be a king as only three of the 51 cards left are Kings. •If the first card was not a king, then the second card is slightly more likely to be a king, as four of the 51 cards left are king
range
the difference between the highest and lowest scores in a distribution
Transpose of a Matrix
the matrix A^t obtained by interchanging rows and columns [6. 4. 24] ^ T= [1. -9. 8] [6. 1] [4. -9] [24 8]
#28- which of the following statements describes a set of data represented in a histogram (outcome:frequency): 1 outcome :10, 2:30, 3:50, 4:30, 5:20, 6:10, 7:10?
the mean is greater than the median. this question requires examity to analyze data in a variety of representations. The mean can be calculated as [10(1)+30(2)+50(3)+30(4)+20(5)+10(6)+10(7)]÷160= 3.5625. the median is the 50th percentile which is 3. The mode is the most frequent value which is 3. The range 7 - 1 equal 6.
Radians
the measure of an angle obtained by dividing the length of the arc by the radius or when the radius is wrapped around the circle.
Median
the middle score in a distribution; half the scores are above it and half are below it
mode
the most frequently occurring score(s) in a distribution
find the amplitude of the function f(x)=-3cosx and use the language of transformations to describe how the graph is related to the parent function y=cosx.
the new function is reflected across the x axis and vertically stretched by a factor of 3.
row space
the set of all linear combinations of the row vectors
sample space
the set of all possible outcomes of a probability experiment
indefinite integral
the set of functions F(x) + C, where C is any real number, such that F(x) is the integral of f(x)...no specified values...
population standard deviation
the square root of the population variance, σ=√[1/NΣ(Xi-μ)², where sigma is from I to N
examples of a random Event
the toss of a coin, throw of a dice and Lottery draws are all examples of random events
lexicographing ordering
the way to set up a truth table PQ- possible conditions, and logical connectors
the chances of rolling a "4"with a dice?
there is only one face with a 4 on it. Total number of outcomes: 6 (there are six faces all together) so the probability is 1/6.
Two vectors are equal if
they have the same magnitude and the same direction and the same position vector
you sell sandwiches. 70% of people choose chicken, the rest choose something else. What is the probability of selling two chicken sandwiches to the next three customers?
this is just like the heads and tails example, but with 70/30 instead of 50/50. .147=.7×.7×.3 the .7 is the probability of each choice we want, call it p. .7²×.3¹ the 2 is the number of choices, call it k. =p^k × .3¹ (the 1 is the # of opposite choices, so H is: n-k
#8- which of the following equations represents the inverse of Y=6x-4/1+3x?
this question causes examinee to perform operations with functions including compositions and inverses. To find the inverse of a function of the form y=f(x), the original equation is rearranged by solving 4 X as a function of Y. y=6x-4/1+3x=>y(1+3x)=6x-4=>y+3xy= 6x-4=>y+4=6x-3xy=> y+4=x(6-3y)=>x=y+4/6-3y exchanging the variables X and Y results in the inverse function f^-1, y=x+4/6-3x review chapter 4 if needed
#19- quadrilateral ABCD has vertices A(-1,-2), B(3,-1), C(2,2), D(-2,1). which of the following is the most descriptive name for quadrilateral ABCD?
this question requires examinee to analyze a two-dimensional figure using a coordinate system. A sketch of the figure created by fighting a four-point suggest that the quadrilateral ABCD is a parallelogram. One way to confirm this is to show that the lines forming one pair of opposite sides are parallel to each other equal in length. The slope of the line AB=-2-(-1)/-1-3=1/4, which equals the slope of line CD, 2-1/2-(-2)=1/4. the length of line AB=√(-1-3)²+(-2-(-1)²)= √(4²+1²)=√17. the slope of side BC equals -1-(-2)/3-2=-3. does ABCD is not a rectangle because the product of the slopes of side AB and BC is ¼×-3≠-1, so the angle between them is not 90°. ABCD is not a rhombus because the length of side AB=√17 does not equal the length of side BC √(3-2)²+(-1-2)²=√1²+3²=√10. if needed, review competency 10
#27- monthly budget rent 800 food 400 utilities 350 car loan 200 gas 100 other 150 in a circle graph represents the data above what is the measure of the central angle of a sector labeled car loan?
this question requires examinee to analyze data in a circle graph. Use proportional thinking and the fact that there are 360 degrees in a circle. Car loans represent $200 of the $2,000 budget or 10%. 10% of 360 degrees is 36 degrees review chapter 14 for more practice
#15- which of the following Expressions gives 2592 cubic inches as an equivalent volume in cubic feet?
this question requires examinee to analyze the use of various units and unit conversion within the customary system. Use dimensional analysis to convert 2592 in³ to cubic feet. 2592in³×1ft³/12×12×12in=2592×1/12³ft³ competency 8 practice
#26- a sum of $2,000 is invested in a savings account. The amount of money in the account in dollars after T years is given by the equation A=2000 eWhat is^.05t. what is the approximate average value of the account over the first two years?
this question requires examinee to apply integration to solve real-world problems. The average value of a continuous function f(x) over an intergal [a b] is 1 /b-a ∫(from b to a) f(x)dx. since the independent variable T represents the number of years the average daily balance over 2 years will be 1/2 of the integral of the function evaluated from 0 to 2: 2000e^.05dt=1000/.05 (e¹-1)=2103 for more practice, review chapter 13
#25 - ∫₀²e^(3x)dx
this question requires examinee to calculate the integral of an exponential function. Let u=3x Then du=3dx, so ⅓du=dx. Rewrite using u and du Let u=3x Find du/dx Rewrite 3/1 Divide 3 by 1 u lower = 3×0, multiply 3×0, u lower = 0 Substitute the upper limit in for x in u=3x u upper=3×2, multiply 3×2, u upper =6 *The values found for u lower and u upper integral u lower- 0 u upper- 6 Rewrite the problem using u, du, and the new limits of integration. ∫₀⁶e^(u)⅓du Since ⅓ is constant with respect to u, move ⅓ out of the integral ⅓∫₀⁶e^(u)du The integral of e^(u) with respect to u is e^(u) ⅓e^(u)]₀⁶ Combine ⅓ and e^(u)]₀⁶ e^(u)]₀⁶ Substitute and simplify Evaluate e^(u) at 6 and at 0 (e⁶-e⁰)/3 Simplify Anything raised to 0 is 1 (e⁶-1×1)/3 Multiply 1×-1 (e⁶-1)/3 review chapter 13 for additional practice
#22- which of the following are the solutions to 2sin²ϴ=cosϴ+1 for 0< ϴ< or equal to 2< or equal to 2pi?
this question requires examining to manipulate trigonometric expressions and equations using techniques such as trigonometric identities. Since sin²ϴ=1-cos²ϴ,2sin²ϴ=cosϴ+1=> 2(1-cos²ϴ)=cosϴ+1=>2cos²ϴ+cosϴ-1=0=> (2cosϴ-1)(cosϴ+1)=0=> cosϴ=½ or cosϴ=-1. thus for 0<ϴ< or equal to 2pi, ϴ=pi/3, 5pi/3, or pi review competency 11 for more practice
#20-the vertices of triangle ABC are A(-5,3), B(2,2), and C(-1,-5). which of the following is the length of the median from vertex B to side AC?
this question requires examity to apply concepts of distance midpoint and slope to classify figures and solve problems in the coordinate plane. The midpoint of side AC where it's median intersects is computed as [(-5+-1)/2, (3+-5)/2] = (-3, -1). the distance from B(2,2) to (-3,-1) is computed as d=√(-3,-2)²+(-1,-2)² = √25+9= √34 review competency 10 for more practice
#14-Which of the following represents the domain of the function f(x)= √2x+3/3x+1?
this question requires the examinee to analyze rational, radical, absolute value and piecewise defined functions in terms of domain, range and asymptotes. Unless otherwise specified, the domain of a function is the range of values for which the function has a real-number value. A rational function must have a nonzero denominator and solving the equation 3x + 1 = 0 yields x= -⅓. Thus this value must be excluded from the domain. The radical expression in the numerator must have a non- argument and solving the inequality 2x+3≥0 yields x≥-3/2. putting these two results together results in -3/2≤X<-⅓ or X>-⅓. The "or" represents the union of the two sets defined by the inequalities, or the union of two intervals. Competency 7 practice.
#30- the heights of adults in a large group are approximately normally distributed with a mean of 65 in. If 20% of the adult fights or less than 62.5 in what is the probability that a randomly-chosen adult from this group will be between 62.5 and 67.5 in tall?
this question requires the examinee to analyze uniform, binomial, and normal probability distributions. A normal distribution is symmetric about the mean. thus if 20% of the Heights or less than 62.5 in (2.5 in from the mean) then 20% of the heights will be greater than 67.5 in (also 25 inches from the mean). Thus 100% - (20%+20%) = 60% and the probability is 60 but the adult will be between 62.5 and 67.5 in tall. Review competency 15 for more practice
#20- vertices of triangle ABC are A(-5,3), B(2,2), and C-1,-5). which of the following is the length of the median from vertex B to side AC?
this question requires the examinee to apply concepts of distance midpoint and slope to classify figures and solve problems in the coordinate plane. The midpoint of side AC where it's median intersects is computed as ((-5+-1)/2, (3+-5)/2)= (-3,-1). The distance from B(2,2) to (-3,-1) is computed as d=√(-3-2)²+(-1-2)²=√25+9=√34. practice competency 10
Matrix Multiplication is not commutative, true or false?
true, AB≠BA
example: a matrix with 3 rows and 5 columns can be added to another Matrix of 3 rows and 5 columns?
true, but it could not be added to a Matrix with 3 rows and 4 columns (The Columns don't match in size).
orthogonal vectors
two vectors whose dot product equals zero
Vector Addition and Subtraction
u-v=u+(-v)= (a-c, b-d)
∫(x+1)³dx=
u=x+1, du=dx... let me see...the derivative of x+1 is...well it is simply 1. So we can have this: ∫(x+1)³dx=∫(x+1)³×1dx ∫(u)³du Then integrate: ∫u³du=(u⁴)/4+C Now put u=x+1 back again: (x+1)⁴/4+C
∫x/(x²+1)dx=
u=x²+1 du=2x, so ½du=xdx ½∫2x(x²+1)dx Rewrite using u, du ½∫1/u du Then integrate: ½∫1/u=½ln(u)+C Now put u=x²+1 back again: ½ln(x²+1)+C
Integration
used to find areas, volumes, Central points, and many useful things. Often it is used to find the area under the graph of a function.
vectors in a rectangular plane
v=(x₂-x₁)i+(y₂-y₁)j
∫x dx
variable x²/2+C
∫x² dx
variable x³/3+C
#6- If p and q are prime numbers and 4/q^3=p^2/50, what is the value of (p+q)?
variables can be isolated by multiplying both sides of the equation by 50q^3, which yields 200=p^2q^3. if p and Q are both Prime then, p^2q^3 is a prime factorization of 200. Since 200=25×8=5^2×2^3 and 5 and 2 are both prime. p must be 5 and q must be 2. so, p+q= 5+2=7, chapter 3
what is ∫x³ dx:(use the power rule)?
we can use the power rule, where n=3, n≠1, ∫[x^(n+1)/n+1]+C =x⁴/4+C
tossing a coin
when a coin is tossed, there are 2 possible outcomes: •heads (H) •tails (T) we say that the probability of the coin Landing H is 1/2 and the probability of the coin Landing T is 1/2
throwing dice
when a single die is thrown (even repeatable) a single throw gives six possible outcomes and sample points: 1,2,3,4,5,6. The probability of any one of them is 1/6.
interval of zero length
when the interval starts and ends at the same place, the result is 0
Basic Counting Principle
when there are "m" ways to do one thing and "n" ways to do another, then there are m×n ways of doing both
replacement
when we put each card back after drawing it the chances don't change, as the events are independent
cosϴ
x/r
∫ln(x)dx
xln(x)-x+C, exponential
sinϴ
y/r
(see diagram) two girls are standing 100 feet apart. They both see a beautiful seagull in the air between them. The angles of elevation from the girls to the bird are 20° and 45° respectively. How high up is the seagull?
y= 26.633 feet 1. tan20°=y/x .363x=y 2. tan45°=y/(100-x) .363x=100-x +1x. +x 1.363x=100 x=73.37 3. y=100-73.37= 26.633feet
natural logarithmic function
y=lnx, has a base e
indirect proof (proof by contradiction)
you temporarily assume that what you are trying to prove is false
z score formula
z = (x - μ)/σ
De Morgan's Rule
~(p • q) :: (~p v ~q) ~(p v q) :: (~p • ~q)
example: you toss a coin three times and it comes up "heads" each time...what is the chance the next toss will also be a "head"?
½
derivative of a square root, √x?
½x^(-½)
180° in radians
π or 3.142 radians
90° in radians
π/2 radians or approximately 1.571
60° in radians
π/3 radians or approximately 1.047
45° in radians
π/4 radians or approximately. 785
30° in radians
π/6 radians or approximately. 524
deck of cards sample point
• 5 of clubs is a sample point • the King of Hearts is a sample Point not just a king since there are 4 Kings
kite (need to see diagram?)
•a quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent •the angles where the two pairs meet are equal •the diagonals, shown as Dash lines above, meet at a right angle. •One of the diagonals bisects (cut equally in half) the other. • each pair is made up of two equal length sides that join up
steps to follow when proving indirectly
•assume the opposite of the conclusion (second half) of the statement. •Proceed as if this is something is true to find the contradiction. •Once there is a contradiction, the original statement is true. • do not use specific examples. use variables so that the contradiction can be generalized.
examples of an event with outcomes
•choosing a "king" from a deck of cards (any of the 4 Kings) is also an event. •Rolling an "even number" (2,4, or 6) is an event.
examples of an event
•getting a tail when tossing a coin is an event •rolling a 5 is an event
examples of things that follow a normal distribution
•heights of people •size of things produced by machines •errors in measurements •blood pressure •marks on a test
p^(k)(1-p)^(n-k)
•p is the probability of each choice we want •k is a number of choices we want •n is a total number of choices
other names for quadrilateral:
•quadrangle •tetragon
mutually exclusive examples
•turning left or right are mutually exclusive (you can't do both at the same time) • heads and tails are mutually exclusive •Kings and aces are mutually exclusive
adding intervals (see diagram in notes?)
ₐ∫^b f(x)dx=ₐ∫^c f(x)dx+ ᵪ∫^b f(x)dx ᵪ=c
cos(π/6)
√(3)/2
sin(3π/4)
√2/2
sin(π/4) or sin(45)
√2/2
cos(π/4)
√2/2 Q1
Sin(π/3) 60
√3/2
sin(2π/3)
√3/2
length of a line
√[(x₁-x₂)²+(y₁-y₂)²]
∫cf(x)dx
∫f(x)dx, multiplication by constant
Sum Rule of Integration, ∫(f+g)dx
∫fdx+∫gdx
Difference Rule of Integration, ∫(f-g)dx
∫fdx-∫gdx
#10- order 1. order 2. order 3 soft drink. 4. 6. 3 large pizza. 1. 2. 1 garlic bread 1. 1. 0 total cost. 19.62. 34.95. 16.50 given the table of orders and total costs above, and that there is a solution to the problem. which of the following Matrix equations could be used to find D, P, & G, the individual prices for a soft drink, a large pizza and garlic bread respectively?
⊹411⊹ ⊹d⊹ = ⊹19.62⊹ ⊹621⊹ ⊹p⊹ = ⊹34.95⊹ ⊹310⊹ ⊹g⊹ = ⊹16.50⊹ Competency 5 this question requires the examinee to solve problems of linear equations or inequalities using a variety of methods. The system of linear equations can be solved by using Matrices. each order can be expressed as a equation with all three equations written with a the variables in the same sequence. The first order is represented by the equation 4d+p+g= 19.62, the 2nd order by 6d+2p+g= 34.95, and 3rd order by 3d+p= 16.50. The Rows of the left-hand Matrix contain the coefficients of d, p, and g for each equation; (411), (621), and (310). the middle matrix contains two variables DPG. The right hand Matrix vertically arranges the constants of the equations.